Theorem fconst2g 3840

Description: A constant function expressed as a cross product.

Assertion

Ref	Expression
fconst2g	|- (B e. C -> (F:A-->{B} <-> F = (A X. {B})))

Proof of Theorem fconst2g

Step	Hyp	Ref	Expression
1		fvconst 3834	. . . . . . . 8 |- ((F:A-->{B} /\ x e. A) -> (F` x) = B)
2	1	adantlr 393	. . . . . . 7 |- (((F:A-->{B} /\ B e. C) /\ x e. A) -> (F` x) = B)
3		fvconst2g 3839	. . . . . . . 8 |- ((B e. C /\ x e. A) -> ((A X. {B})` x) = B)
4	3	adantll 392	. . . . . . 7 |- (((F:A-->{B} /\ B e. C) /\ x e. A) -> ((A X. {B})` x) = B)
5	2, 4	eqtr4d 1508	. . . . . 6 |- (((F:A-->{B} /\ B e. C) /\ x e. A) -> (F` x) = ((A X. {B})` x))
6	5	r19.21aiva 1712	. . . . 5 |- ((F:A-->{B} /\ B e. C) -> A.x e. A (F` x) = ((A X. {B})` x))
7		eqid 1474	. . . . 5 |- A = A
8	6, 7	jctil 292	. . . 4 |- ((F:A-->{B} /\ B e. C) -> (A = A /\ A.x e. A (F` x) = ((A X. {B})` x)))
9		eqfnfv 3792	. . . . 5 |- ((F Fn A /\ (A X. {B}) Fn A) -> (F = (A X. {B}) <-> (A = A /\ A.x e. A (F` x) = ((A X. {B})` x))))
10		ffn 3623	. . . . 5 |- (F:A-->{B} -> F Fn A)
11		fconstg 3654	. . . . . 6 |- (B e. C -> (A X. {B}):A-->{B})
12		ffn 3623	. . . . . 6 |- ((A X. {B}):A-->{B} -> (A X. {B}) Fn A)
13	11, 12	syl 10	. . . . 5 |- (B e. C -> (A X. {B}) Fn A)
14	9, 10, 13	syl2an 454	. . . 4 |- ((F:A-->{B} /\ B e. C) -> (F = (A X. {B}) <-> (A = A /\ A.x e. A (F` x) = ((A X. {B})` x))))
15	8, 14	mpbird 196	. . 3 |- ((F:A-->{B} /\ B e. C) -> F = (A X. {B}))
16	15	expcom 374	. 2 |- (B e. C -> (F:A-->{B} -> F = (A X. {B})))
17		feq1 3616	. . 3 |- (F = (A X. {B}) -> (F:A-->{B} <-> (A X. {B}):A-->{B}))
18	17, 11	syl5cbir 211	. 2 |- (B e. C -> (F = (A X. {B}) -> F:A-->{B}))
19	16, 18	impbid 515	1 |- (B e. C -> (F:A-->{B} <-> F = (A X. {B})))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 955   e. wcel 957  A.wral 1643  {csn 2406   X. cxp 3164   Fn wfn 3173  -->wf 3174  ` cfv 3178

This theorem is referenced by:  fconst2 3842  fconst5 3843

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-fv 3194

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